\documentclass[onecolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
% Some other (several out of many) possibilities
%\documentclass[preprint,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B

\usepackage{amsmath,amssymb,amsfonts,amstext} % extended math symbols etc
\usepackage{graphicx}% Include figure files
\usepackage{bm}% bold math

\newcommand{\comment}[1]{\begin{tabular}{||p{\textwidth}}\small \textbf{Comment:} \it #1 \end{tabular}}
\def\code#1{{\tt #1}}


\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\vecj}{\Vec{j}}
\newcommand{\veck}{\Vec{k}}
\newcommand{\vecr}{\Vec{r}}
\newcommand{\vecx}{\Vec{x}}
\newcommand{\vecy}{\Vec{y}}

% hyperref package should always be loaded as the very last one
% to be sure that it has the last word ...
\usepackage[
  pdftex,
  pdftitle={Notes on Implementing Density Functional Theory of Hard Sphere Charged Liquids in 1D and 3D},
  pdfauthor={Dmitry~A. Karpeev, Matthew~G. Knepley},
  pdfpagemode={UseOutlines},
  bookmarks, bookmarksopen,bookmarksnumbered={True},
  colorlinks, linkcolor={blue},citecolor={blue},urlcolor={blue}
]{hyperref}

\renewcommand{\baselinestretch}{2}

\begin{document}

\title{Notes on Implementing Density Functional Theory of Hard Sphere Charged Liquids in 1D and 3D}

\author{Dmitry~A. Karpeev}
\affiliation{Mathematics and Computer Science Division\\Argonne National Laboratory}
\email{karpeev@mcs.anl.gov}
\author{Matthew~G. Knepley}
\affiliation{Computation Institute\\University of Chicago}
\email{knepley@ci.uchicago.edu}


\date{\today}


%\pacs{Valid PACS appear here}

\maketitle
%\keywords{Suggested keywords} % Use showkeys class option if keyword display desired
\section{1D Window Functions and their Fourier Transforms}
\subsection{Auxiliary Fourier Transforms}
Consider the following family of Fourier integrals (moments) depending on a natural index $n = 0,1, \dots$,
a positive real parameter $0 < R < \infty$ and a real variable $k \in \R$:
\begin{equation}
\label{eq:I-integrals}
        I_{R,n}(k) = \frac{1}{\sqrt{2\pi}}\int_{-R}^{R} x^n e^{-ikx} dx.
\end{equation}
These exist in the usual (Lebesgue or even Riemann) sense and can be easily evaluated by integrating by
parts as follows.
\paragraph{{\bf Base case}: $n = 0$}.
\begin{equation}
I_{R,0}(k) = \frac{1}{\sqrt{2\pi}}\int_{-R}^{R} e^{-ikx} dx = \frac{R}{\sqrt{2 \pi}} \frac{\sin{(kR)}}{kR} =
\frac{R}{\sqrt{2 \pi}} \text{sinc}\,(kR),
\end{equation}
where $\text{sinc}$ denotes the {\it unnormalized sinc} function.
\paragraph{{\bf Generic case}}.
Given the base case the other moments are easily computed recursively as follows:
\begin{multline}
        I_{R,n}(k) = \frac{1}{\sqrt{2\pi}}\int_{-R}^{R} x^n e^{-ikx} dx = 
                \frac{1}{\sqrt{2\pi}}\int_{-R}^{R} \frac{i}{k} \frac{d\left(x^n e^{-ikx}\right)}{dx} dx  - 
                n\frac{1}{\sqrt{2\pi}}\int_{-R}^{R} x^{n-1} e^{-ikx} dx = \\
                \frac{i}{k\sqrt{2 \pi}} R^n \left[e^{-ikR} + (-1)^n e^{ikR}\right] - n I_{R,n-1}(k).
\end{multline}
\bibliography{dft-fft}

\end{document}
